Theoretical and Experimental DNA Computation (Natural ...

4.9 P-RAM Simulation 91
algorithm may be emulated in DNA. The P-RAM is a widely used model in the
design of efficient parallel algorithms, especially by the international community
of theorists working in the field. There are literally thousands of published
algorithms within this model which, often employing ingenious combinatorics,
are remarkably efficient. One of the attractions of our approach is that we can
call upon this vast published expertise. The immediate difficulty however is
that it is not clear how to directly emulate the P-RAM in DNA without invoking
great intricacy and unrealistic bench chemistry. On the other hand, we
have been able to describe how a Boolean circuit may be efficiently emulated
in DNA. Although both models are universal (that is, are Turing-complete),
it is in general very difficult to “program” a Boolean circuit (that is, to design
such a circuit to solve a specific problem) without a great degree of low
level intricacy and opacity. The reverse is true for the P-RAM which yields
a natural, high-level, programming environment. The benefit of our methodology
is that we are able to describe a general method for compiling P-RAM
algorithms into Boolean Combinational Logic Networks.
Overall, then, we believe that we have a theoretical model which is easy
to program, may call upon a vast literature of efficient parallel algorithmic
design, and which (through Boolean circuit emulation) has the prospect of
emulation in DNA using relatively simple and clean processes. Moreover, and
very importantly, there is only a logarithmic loss in efficiency in the emulation
process. We now outline, in greater technical detail, how the emulation works.
The work presented in [8] gives a simulation of Boolean networks by DNA
algorithms that requires parallel time comparable to the network depth and
volume comparable to the network size. This follows the work of Ogihara and
Ray [113], in which they describe a Boolean circuit model that, within our
strong model of DNA computation [10], runs in parallel time proportionate
to the circuit size. Although we point out in [8] that a simple modification
to Ogihara and Ray’s simulation would achieve the same time complexity of
our simulation, we believe that the biological implementation of our model is
more straight forward. We justify this remark later.
The crucial extension of our simulation is presented in [9]: this presents
a full detailed translation from the high-level P-RAM algorithms into DNA,
the translation being accomplished by using a Boolean network representation
of the P-RAM algorithm as an intermediate stage. In total this translation
achieves the following performance: if A is a P-RAM algorithm using P (n)
processors, taking S(n) memory space, and taking T (n) time, then the total
computation time of the DNA algorithm is bounded by O(T (n)logS(n)),
and the total volume of DNA used is O(P (n)T (n)S(n)logS(n)). As a consequence
this gives a direct translation from any NC algorithm into an effective
DNA algorithm. The simulation presented thereby moves the purely theoretical
“in principle” result that DNA can realize NC “efficiently” into the
realm where such algorithms can be realized in DNA by a practical, universal
translation process. In [9] a formal definition of the low-level instruction set
available to each processor in the P-RAM is specified: this provides standard